Here's an idea for a better problem:

Solve the following:

x^3 = 27

y^3 = 21

z^4 = 85

My rationale is that there will be a huge time advantage for the student who works out the solutions by using roots, and a visual "hint" that there might be a general solution.

But I'm of two minds about it. I love manipulating expressions by hand. It's a relaxing hobby. But it limits the choice of problems that can be solved, which in turn narrows the range of things that can be taught, and even creates a false sense of what is possible in math. And it doesn't reflect how math is used by most people, i.e., with a computer.

I'd rather incorporate more computers into the math curriculum, and maybe merge math and programming into a single subject.

x^3 = 27 (calculating the volume of a cube)

is an application of the equation of

x^y = z (potentiate a number)

which is an application of the equation

f(x) = y (apply a function)

The solution is simply

x = f-inverse(y)

if f is invertible.

As this simple example shows math on high school level can not splitted up into geometry, algebra, calculus, etc. Understanding one of these areas helps to understand the others and vice versa. If you want to master one of them you have to master all of them at the same time, with the same speed, parallel.

x^3 = real number

That's the simplest solution. It works in every case. To me the answer is not important. The methodology is important. Giving a counter example is very much a different type of problem. Just about any method is valid in that type of problem.

Passing a class should mean more than I got a lot of answers correct. It should mean an understanding of the material. A college algebra student who solves x^3=27 in the aforementioned manner is lacking a fundamental understanding of the material. Now a third grader who reasons thusly, well that is impressive. The goal is not the right answer. It a demonstration of understanding and abilit appropriate to the level of the course.

> Just about any method is valid in that type of problem.

Why is that not true for other types of problems?

> Passing a class should mean more than I got a lot of answers correct.

I agree. But you shouldn't penalise the student if the exam question is poorly framed (and we all make such mistakes). Just take a note for later and don't make the mistake again.

It's interesting reading all the replies I've gotten. It's nice to see other peoples' perspectives. Including yours.

As you stated I would not give x^3=27 as a problem in college algebra. It's a fine line and I suspect that we mostly agree except on one part.

As a grader I've given full credit for the wrong answer and no credit for the right answer.

I think I am misunderstanding something, because as far as I can tell, integration by parts where one of the parts is 1 is literally useless.

du = 1/x and v = x

integral ln(x) = x ln(x) - integral 1/x times x

integral ln(x) = x ln(x) - x + C

I could give integral arctan(x) but with the advent of computer algebra systems I'm mostly interested in them knowing the basic examples and to not burden them on a test with something more complicated.

EDIT: The derivative of 1/x is not ln(x) as you stated. You got it backwards and my guess is that is the source of your confusion.

I agree, but testing for understanding (as opposed to Socratically probing for it) is more time consuming, complex, and difficult than just testing for correct answers.

To stay within a given test workload, students would have to take far fewer tests. Obviously not the direction the educational system is trending these days. Which is a shame.

You get 1 mark for the right answer and 2 marks for working. Problem solved. The question doesn't have to change at all. This is how I remember mark schemes working in the general case

I'm not sure I get what you mean by "working". Can you expand that a bit?

Not just true in mathematics but in engineering as well. I was taking an engineering course in my sophomore year where I had a system of equations in 3-4 variables. I spent forever trying to solve it (analytically), and failed. So did most of the class. The next lecture, the professor showed us how to do it. A mixture of plots, etc reduced the solution space and the rest was trivial. He also said "You could just use the solver in your calculator/MATLAB".

I wasn't satisfied with his answer. It felt like cheating. I didn't learn the cool way to do things.

But in the real world, if you can get the solution this way, it's perfectly valid. As long as you can confirm that you found a/the solution (trivial to do).

With the x^3=27 answer, it is the onus of the instructor to specify explicitly that "guessing is not allowed". Why? Because as others have mentioned, it is totally appropriate in mathematical circles to guess a solution. Much work in mathematics is done that way.

In various classes (mathematics/engineering/physics), I've both utilized non-standard ways to solve problems on tests, and have seen it done by students on tests I grade. This is to be encouraged. Especially because this is what mathematicians/physicists love to do in their real work.

If your goal is to ensure they understand cube roots, either make a problem that is hard to guess (e.g. x^3 = 24), or be explicit about it. Even with x^3 = 24, if they use Newton's method, that should be graded correct.

Guessing is not a method of solving. It is a method of finding counterexamples. Two different types of problems.

I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.

Believe it or not, proving theorems is not the goal of many mathematicians. I'm simplifying a bit, but read Freeman Dyson's essay on Birds and Frogs. Essentially "problem solvers" vs "theory builders". While problem solvers often do end up proving theorems, it is not their main goal. If they can "guess" a solution, they are done. It is publishable.

Go to the field of combinatorics, and you'll find it is full of guesswork.

>I accept any mathematically valid method of solving a problem. Mathematically method means, method that works even if I trivially change it by using different numbers.

Sorry, but many mathematicians disagree with you. Solving a problem is finding a solution (provided you have a means to verify correctness). It doesn't matter if you merely guessed it.

I have the book by Stanley on combinatorics. There are not results of the form: I guessed A is the answer and it's right. Let's move on. This is does not happen. When one notices something is a solution the mathematician always wonders why. A mathematician wonders what underlying structure there is. Never is one satisfied by a guess.

If someone found a counterexample to the Riemann Hypothesis the first question would be, why is this number a counterexample? What caused the obstruction? You could problem publish a paper that just said, A is a counterexample to the Riemann Hypothesis. But you could not publish a paper that said, I guessed A is a solution to B and it turns out I was right.

This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.

Any method of finding a counterexample is accepted. Guessing a solution is not.

EDIT: The paper linked to was published because it was a counterexample to a famous conjecture.